$B$ is again an axis-parallel rectangle in the $(x,y)$ plane.
hint
After finding the restriction of $f$ and multiplying with the Jacobian you will arrive at a double integral like this (assuming you chose a parametrization like the one in the previous exercise):
$$\int_{0}^{1}\, \left(
\int_{0}^{1} \frac{v}{1+uv}\,\, \mathrm du \, \right)\mathrm dv\,
$$
hint
For the innermost integral, use the method of substitution by choosing the inner function as $g(u)=1+uv$ and the outer function as $f(u)=\displaystyle{\frac 1u}\,.$
answer
$$\int_B \frac{y}{1+xy} \;\mathrm d\mu=2\ln 2-1$$
Exercise 2: Parametrization and the Plane Integral. By Hand
In the $(x,y)$ plane the point $\displaystyle{P_0=\left(2\,,1\right)\,}$ is given along with the set of points
Make a preliminary sketch of $B\,$ and state a parametric representation $\mr(u,v)$ of $B\,$ with suitable intervals for $u$ and $v\,.$ Determine two numbers $u_0$ and $v_0$ such that $\mr(u_0,v_0)=P_0\,.$
With this parametrization, the values $u_0=2$ and $v_0=\frac12$ will fulfill $\mathbf r(2,\frac12)=P_0$.
B
Make an illustration of $B$ using Maple where you from $P_0$ draw the tangent vectors $\mr’_u(u_0,v_0)$ and $\mr’_v(u_0,v_0)\,.$ Determine the area of the parallelogram spanned by the tangent vectors.
hint
For the area compute the absolute value of the determinant of the two vectors.
C
Determine the Jacobian function corresponding to $\mr(u,v)$, and compute the plane integral
A function $\,f:\reel^2\rightarrow \reel\,$ is given by
$$\,\displaystyle{f(x,y)=x^2-y^2}\,.$$
For a given point in the $\,(x,y)$ plane $\,\varrho\,$ denotes the point’s absolute value (the distance from the point to the origin). Similarly, $\,\varphi\,$ denotes the argument of the point (the angle between the $x$ axis and the position vector of the point given a positive sign when turned counter-clockwise).
A set of points $B$ is in polar coordinates described by
The tangent vectors $\,\mr’_u(u_0,v_0)\,$ and $\,\mr’_v(u_0,v_0)\,$ drawn from $\,P_0\,$ span a parallelogram $\,\mathcal P\,.$
State a parametric representation of $\,\mathcal P\,.$
C
Make a Maple illustration that contains all of the following:
$F_{\mr}\,$
$\mr’_u(u_0,v_0)\,$ and $\,\mr’_v(u_0,v_0)\,$
The normal vector $\,\mathbf n(u_0,v_0)=\mr’_u(u_0,v_0)\times \mr’_v(u_0,v_0)\,$
$\,\mathcal P$
hint
Find inspiration for the plotting syntaxes in today’s Maple demo.
D
Compute the area of $\,\mathcal P\,.$
hint
The area is found as the length of the normal vector.
hint
$$\mathbf N(u,v)=(\sin v, -\cos v,u)$$
answer
$$\sqrt 2$$
E
Clarify to a fellow student how the formulas for the Jacobians are different for plane integrals and for surface integrals.
F
Compute the Jacobian function corresponding to $\mr(u,v)$, and compute the area of $F_{\mr}\,.$
hint
For the area, compute the surface integral of the Jacobian without any function involved.
answer
$$(2\sqrt 5-\ln(\sqrt 5-2))\pi \approx 18.6$$
Exercise 5: Cylindrical Surfaces
A cylindrical surface is a surface that is vertically perpendicular to a so-called directrix in the $(x,y)$ plane. Despite the name, a cylindrical surface does not have to have the form of a cylinder. For the cylindrical surface to be well-defined, a $z$-interval must be given for all points $(x,y)$ on the directrix.
A cylindrical surface $\mathcal C_1$ is given by the following information:
State a parametric representation of $\mathcal C_1\,.$
hint
First make a parametrization of the directrix. This defines the bottom (or a cut-through) of the cylindrical surface. Then add the third coordinate from the given information.
hint
The directrix is a circle centred at $(1,0)$ with a radius of $1$.
We are given a mass density distribution function:
$$f(x,y)=1\,.$$
A
Compute the mass of the set.
hint
The mathematical plane integral that we have worked with all day has an interpretation in physics depending on what the function $f$ that we integrate over represents. When it represents mass density, then the computed plane integral is the total mass of the set.
answer
$$M=\frac{15}{4}$$
B
A physics question: Discuss with a fellow student what SI units $f$ and $M$ would have.
answer
A mass density function on a surface would in SI units be expressed in kilograms-per-square-metre, $\mathrm{kg/m^2}$.
A mass would simply be meaured in kilograms, $\mathrm{kg}$.
C
Determine the center of mass of the set.
hint
See the centre-of-mass formula and approach in today’s Maple demo.
answer
$$\left(\frac{124}{75},\frac{254}{105}\right)$$
D
Determine the mass and the center of mass when the mass density is $f(x,y)=x^2.$
